In this paper we discuss the problem

We show that for c > 0 there exists a positive constant μ* such that (*)μ possesses at least one solution if μ ∈ (0, μ*) and no solutions if μ > μ*. Furthermore, there exists a positive constant μ**≦μ* such that (*)μ possesses at least two solutions if μ∈(0, μ**), 2<N<6. For N≧6, μ∈(0,μ**), we show that problem (*)μ possesses a unique solution if f(x) is radial with f′(r) < 0(r = |x|).

The effect of the presence of singularities on the method of stationary phase for a certain integral of two variables is examined. The singularities lie on two straight lines whose positions depend upon parameters. For certain values of the parameters the straight lines can pass through the point of stationary phase.

The asymptotic development, which is uniform with respect to the parameters, is determined. It is found that it can be expressed in terms of a function which, in certain circumstances, reduces to a Fresnel integral. The main properties of this function are derived in an appendix.

Explicit formulae are given for the dominant asymptotic terms in the cases when the point of stationary phase is approached by (i) neither line, (ii) only one of the lines, (iii) both lines.

Let G be a finite group, and let cs(G) be the set of conjugacy class sizes of G. Recalling that an element g of G is called a vanishing element if there exists an irreducible character of G taking the value 0 on g, we consider one particular subset of cs(G), namely, the set vcs(G) whose elements are the conjugacy class sizes of the vanishing elements of G. Motivated by the results inBianchi et al. (2020, J. Group Theory, 23, 79–83), we describe the class of the finite groups G such that vcs(G) consists of a single element under the assumption that G is supersolvable or G has a normal Sylow 2-subgroup (in particular, groups of odd order are covered). As a particular case, we also get a characterization of finite groups having a single vanishing conjugacy class size which is either a prime power or square-free.

The conjecture of Muller-Pfeiffer [4] concerning the oscillation behaviour of the differential equation (–l)n(p(x)y(n))(n) + q(x)y = 0 is proved, and a similar conjecture concerning the more general differential equation ∑nk=0(−l)k(Pk(x)y(k)(k + q(x)y= 0 is formulated.

Following pioneering work by Fan and Slemrod, who studied the effect of artificial viscosity terms, we consider the system of conservation laws arising in liquid–vapour phase dynamics with physical viscosity and capillarity effects taken into account. Following Dafermos, we consider self-similar solutions to the Riemann problem and establish uniform total variation bounds, allowing us to deduce new existence results. Our analysis covers both the hyperbolic and the hyperbolic–elliptic regimes and apply to arbitrarily large Riemann data.

The proofs rely on a new technique of reduction to two coupled scalar equations associated with the two wave fans of the system. Strong $L^1$ convergence to a weak solution of bounded variation is established in the hyperbolic regime, while in the hyperbolic–elliptic regime a stationary singularity near the axis separating the two wave fans, or more generally an almost-stationary oscillating wave pattern (of thickness depending upon the capillarity–viscosity ratio), is observed and the solution may not have globally bounded variation.

Let W be a semigroup with W\W2 non-empty, such that if ρ is a congruence on W with xpy for all x, y= W\W2, then zpw for all z, w= W2. We prove that the lattice of congruences on W is directly indecomposable, and conclude that a direct product of cyclic semigroups, with at least two non-group direct factors, has a directly indecomposable lattice of congruences. We find that the lattice of congruences on a direct product S1×S2×V of two non-trivial cyclic semigroups S1 and S2, one not being a group, and any other semigroup V, is not lower semimodular, and hence, not modular. We then prove that any finite ideal extension of a group by a nil semigroup has an upper semimodular lattice of congruences, and conclude that a finite direct product of finite cyclic semigroups has an upper semimodular lattice of congruences.

In this paper we introduce the concept of fractional powers of a pair of operators between two Banach spaces. The operators need not be closed, but form a closed pair. The properties of the fractional powers are studied. An application of the theory is briefly discussed.

We prove nonlinear stability of planar shock fronts for certain relaxation systems in two spatial dimensions. If the subcharacteristic condition is assumed and the initial perturbation is sufficiently small and the mass carried by the perturbations is not necessarily finite, then the solution converges to a shifted planar shock front solution as time t ↑ ∞. The asymptotic phase shift of shock fronts is, in general, non-zero and governed by a similarity solution to the heat equation. The asymptotic decay rate to the shock front is proved to be t−1/4 in L∞(R2) without imposing extra decay rates in space for the initial perturbations. The proofs are based on an elementary weighted energy analysis to the error equation.

Let 𝕀 denote an imaginary quadratic field or the field ℚ of rational numbers and let ℤ𝕀 denote its ring of integers. We shall prove a new explicit Baker-type lower bound for a ℤ𝕀-linear form in the numbers 1, eα1 , . . . , eαm, m ⩾ 2, where α0 = 0, α1, . . . , αm are m + 1 different numbers from the field 𝕀. Our work gives substantial improvements on the existing explicit versions of Baker’s work about exponential values at rational points. In particular, dependencies on m are improved.

In this paper, we consider a semi-classical version of the nonhomogeneous heat equation with singular time-dependent coefficients on the lattice $\hbar \mathbb {Z}^n$. We establish the well-posedness of such Cauchy problems in the classical sense when regular coefficients are considered, and analyse how the notion of very weak solution adapts in such equations when distributional coefficients are regarded. We prove the well-posedness of both the classical and the very weak solution in the weighted spaces $\ell ^{2}_{s}(\hbar \mathbb {Z}^n)$, $s \in \mathbb {R}$, which is enough to prove the well-posedness in the space of tempered distributions $\mathcal {S}'(\hbar \mathbb {Z}^n)$. Notably, when $s=0$, we show that for $\hbar \rightarrow 0$, the classical (resp. very weak) solution of the heat equation in the Euclidean setting $\mathbb {R}^n$ is recaptured by the classical (resp. very weak) solution of it in the semi-classical setting $\hbar \mathbb {Z}^n$.

The main aim of this paper is to introduce an appropriate dyadic one-sided maximal operator , smaller than the one-sided Hardy–Littlewood maximal operator M+ but such that it controls M+ in a similar way to how the usual dyadic maximal operator controls the Hardy-Littlewood maximal operator. We characterize the weighted inequalities for this dyadic one-sided maximal operator.

The middle cells are present throughout the whole length of the spinal cord. They are situated in the middle region of the grey matter, between the free anterior and free posterior cornua; but they sometimes also extend into the regions usually occupied by the anterior cornual cells, by the intermedio-lateral tract or by Clarke's column. The small cells in the base of the anterior horn (scattered cells) cannot be sharply separated from the middle cells, nor can most of the small cells about the base of the free posterior horn.

Although some of the middle cells may be found in all this wide area of distribution at practically any level of the cord, there are certain arrangements of cells which may be looked upon as typical of each segment, or at least of each region of the cord. In the upper cervical region they are not on the whole very abundant (this differs from Waldeyer's account), but are best developed in the central and para-central fields. In the cervical enlargement they are much more numerous, particularly within a broad band extending from the formatio reticularis to about the anterior grey commissure. Throughout the dorsal region they are again comparatively few in number, and may be scattered irregularly; but small groups are often found, most commonly in the central area, about on a level with the central canal, and in the post-central area, between Clarke's column and the reticular group of the intermedio-lateral tract. In the lumbar segments the middle cells are abundant, particularly centrally and in the base of the anterior horn. Their field extends anteriorly, until in the lower sacral region they come to be found over the whole area of the anterior cornua in addition to their more usual situations.

At all levels, as studied in serial sections, the distribution of the middle cells is seemingly erratic and casual. No regular plan of arrangement can be made out, and there is most certainly no segmentation (as Argutinski described) like that so well seen in the intermedio-lateral tract. The cells are sometimes just dotted here and there singly; sometimes they are scattered fairly thickly and evenly over the whole or part of the regions they are to be found in; or, again, they may occur more thickly in one part, or be aggregated into a distinct little cell group or nest, but even then these cell groups are seldom so closely packed as are the cell groups of the intermedio-lateral tract. The duration of any one of these types of cell arrangement is inconstant; and although there are levels where cells seem for a bit to be almost persistent in one place, this much can be stated as a general rule, that no middle cell group lasts through more than a very few serial sections. If traced further, the group is found either to shift to some other area or to die out altogether. Occasionally there appears to be a variation in number of cells parallel with the oscillations of the intermedio-lateral tract, but closer study shows that this is by no means absolute, and is probably only a local accidental variation.

Without expressing any opinion as regards function, I am inclined to divide the cells I have described collectively as middle cells into three groups, basing this division merely on the distribution and arrangement of the cells and on the microscopic appearances of the cell bodies. These divisions are, however, not very sharply defined either as regards the character of the individual cells or in the cell distribution. Still they seem to warrant such a division being made, and to suggest at least that the cells of the three groups may be functionally different. The groups I would make are—

(1) The middle cells proper, or central cells, occurring chiefly in the central area of the grey matter. These cells are of medium size and very similar to those of the apical group of the intermedio-lateral tract, multipolar, polygonal or rounded polygonal in outline, with a comparatively large nucleus and a few chromatic granules round it in the cell substance. Sometimes scattered, but more often present as a small clump of cells. With them may be included the cells in the para-central area, which, although sometimes forming a distinct and separate aggregation of cells, can mostly not be separated off from the central cells. Many of the cells occurring in the reticular area at levels where the reticular group of the intermediolateral tract is not present (especially in the lower cervical region) may probably also be included here.

(2) Anterior central cells in the base of the anterior horn—one of Waldeyer's “scattered” cell groups. As this name implies, these are often simply scattered over the area in question, but sometimes they are gathered into small cell-nests, but rarely very compact ones. Although there is no sharp boundary between them posteriorly and the central cells, they are in the main larger cells, and are further distinguished by their shape. They are not so often definitely multipolar and of approximately equal diameter in different directions, but more frequently appear to be bipolar, with longdrawn-out processes. The axis of elongation may be variously oriented. With these would fall to be included the middle cells in the external central area of the lateral enlargements, particularly in the lumbo-sacral cord. Possibly also some of the para-central cells should be classed with this group, and not with the previous one.

(3) Post-central cells. These lie in the area between Clarke's column and the formatio-reticularis, or in the corresponding region of the grey matter at level where Clarke's column is unrepresented. As with the anterior central cell, there is no sharp boundary between this series and the central middle cells. They are often continuous, or the cells may lie betwixt and between the two areas. But although many of the posterior cells may be as large, they are distinguished on the whole by being smaller in size than the central cells and less definitely polygonal, more rounded in outline. They are generally present either as a small clump or as a band of cells on the outer and posterior aspects of Clarke's column. Posteriorly, it is very difficult to separate this group from the posteriorbasal and posterior-marginal cells of Waldeyer. With this group may be included cells present in the area of Clarke's column, particularly when that column is absent or only represented by occasional cells. Also some of the small cells found in the reticular formation, especially those lying between (vertically) the nuclei of the reticular group of the intermedio-lateral tract.

These subdivisions may require modification from subsequent investigations, particularly by tracing the course and termination of the cell processes. I have attempted to do this by means of Cajal's silver impregnation methods, but, so far, have not succeeded. All I can affirm is that the fibres arising from the middle cells run in a variety of directions to begin with; but as some could be traced far enough to be seen doubling more or less sharply upon themselves, this gives no real clue to their destination. In no case could I follow any one to a termination. In some cells them was observed an endo-cellular fibrillary network, similar to that in the large motor cells of the anterior horn. In this particular cord the following figures give approximate average diameter of the cells of the different groups:—

In conclusion, I have to express my thanks to Dr Alexander Bruce for suggesting this subject for a thesis; to Dr Theodore Shennan, in whose laboratory the preparation of the sections was carried out; to Mr Henry Wade for the use of micro-photographic apparatus; and lastly to the Carnegie Trust, since this thesis is part of work done under the terms of a grant from the Trust for original research.

Let M be a closed 4-manifold with a free circle action. If the orbit manifold N3 satisfies an appropriate fibering condition, then we show how to represent a cone in H2(M; ℝ) by symplectic forms. This generalizes earlier constructions by Thurston, Bouyakoub and Fernández et al. In the case that M is the product 4-manifold S1 × N, our construction complements our previous results and allows us to determine completely the symplectic cone of such 4-manifolds.

In this paper we look for closed geodesies on a noncomplete Riemannian manifold ℳ. We prove that if ℳ has convex boundary, then there exists at least one closed nonconstant geodesic on it.

The Chair which I have been elected to occupy, in succession to Professor Darwin, is associated with the name of a great scholar of our fathers' generation, Peter Guthrie Tait. This name has been familiar to me from the time when I first began to study mathematical physics. At that time Felix Klein was the leading figure in a group of outstanding mathematicians at Göttingen, amongst them Hilbert and Minkowski. I remember how Klein, ever eager to link physics with mathematics, missed no opportunity of pointing out to us students the importance of studying carefully the celebrated Treatise on Natural Philosophy of Thomson and Tait, which became a sort of Bible of mathematical science for us.

Some time ago, in a paper published by the Royal Anthropological Institute, I applied a Mendelian analysis to that part of the observations made by the late Dr Beddoe (1) which refers to the colour of the hair. In that paper (2) I showed that these observations obeyed in a highly remarkable degree the law referred to, and that this result held from the north of Scotland, through the whole of England, Ireland, France, and Germany, to the south of Italy. At that time I was unable to make any application to the observations on eye colour also published in the same work, but I have now succeeded in completing the analysis.

A characteristic feature of the plateau of Cambrian Limestone in the neighbourhood of Inchnadamff is the occurrence in it of swallow-holes, caves, and subterranean channels which are intimately associated with the geological history of the region. The valley of Allt nan Uamh (Burn of the Caves), locally known as the Coldstream Burn, furnishes striking examples of these phenomena. One of the caves in this valley yielded an interesting succession of deposits, from which were collected abundant remains of mammals and birds. The discovery of bones of the Northern Lynx, the Arctic Lemming, and the Northern Vole among these relics, and the collateral evidence of the materials forming some of these layers, seem to link the early history of this bone-cave with late glacial time, or at least with a period before the final disappearance of local glaciers in that region.

It is now thirty-two years since I last brought this subject before the Royal Society of Edinburgh. Since then nothing has been done to remedy the imperfections which were then shown to characterise the thermometer screens used throughout the country. In some respects these imperfections may seem to be of small importance. Yet in these days when accuracy of measurement is being more and more insisted on, and particularly now when reconstruction is everywhere in vogue, it does seem desirable that apparatus largely used by thousands of observers should be as perfect as it is possible to make it.

The asymptotic behaviour of semigroups of nonlinear contractions which have a set of fixed points containing a ball of finite codimension is studied. It is shown that the ω-limit sets of such semigroups are finite dimensional tori, and that an analogue of the classical Kronecker-Weil theorem holds for such semigroups.

We completely classify non-degenerate surfaces with parallel second fundamental form in three-dimensional Riemannian and Lorentzian reducible spaces.